This is a short guide to adapting existing algorithms and problem instances for running an experiment using CAISEr. In this document, we cover:

- Definition of instance lists
- Adaptation of existing algorithms
- Some stuff you can do with the results

A general description of the CAISE methodology is available in our papers: - F. Campelo, F. Takahashi, *Sample size estimation for power and accuracy in the experimental comparison of algorithms*. Journal of Heuristics 25(2):305-338, 2019.]

- F. Campelo, E. Wanner, *Sample size calculations for the experimental comparison of multiple algorithms on multiple problem instances*. Submitted, Journal of Heuristics, 2019.

As stated in the documentation of both `run_experiment()`

and `calc_nreps()`

, each instance must be a named list containing all relevant parameters that define the problem instance. This list must contain at least the field `instance$FUN`

, with the name of the problem instance function, that is, a routine that calculates \(y = f(x)\). If the instance requires additional parameters, these must also be provided as named fields. Each instance can also have an `alias`

, a unique name to distinguish it from other instances. If no alias is provided, the name of the function (`instance$FUN`

) is used as the instance ID.

The `instances`

parameter for `run_experiment()`

is simply a list or vector of these instance lists.

To illustrate how to adapt existing implementations to this structure, we assume that we are interested in comparing four multiobjective optimization algorithms for a (hypothetical) problem class represented by problems UF1 - UF7 (in dimensions between 10 and 40) from package smoof. For this implementation to work with the `MOEADr::moead()`

routine (see next section) some manipulation is necessary, but the instance list in this case is simply a list with each element containing the name of the routine as field `$FUN`

(since all function names are different, no need for aliases).

```
suppressPackageStartupMessages(library(smoof))
suppressPackageStartupMessages(library(MOEADr))
suppressPackageStartupMessages(library(CAISEr))
### Build function names (instances: UF1 - UF7, dimensions 10 - 40)
fname <- paste0("UF_", 1:7)
dims <- c(10:40)
allfuns <- expand.grid(fname, dims, stringsAsFactors = FALSE)
# Assemble instances list
instances <- vector(nrow(allfuns), mode = "list")
for (i in 1:length(instances)){
instances[[i]]$FUN <- paste0(allfuns[i,1], "_", allfuns[i,2])
}
### Build the functions listed above (so that they can be properly used)
for (i in 1:nrow(allfuns)){
assign(x = instances[[i]]$FUN,
value = MOEADr::make_vectorized_smoof(prob.name = "UF",
dimensions = allfuns[i, 2],
id = as.numeric(strsplit(allfuns[i, 1], "_")[[1]][2])))
}
```

We will use the MOEA/D implementation available in the MOEADr package as our base algorithm, and assume that we are interested in comparing the performance of four versions of this algorithm: the two versions described in the original MOEA/D paper, the one described in the MOEA/D-DE paper, and a variation of the MOEA/D-DE configuration that uses PBI as the aggregation function and calculates neighborhoods based on the space of parameters instead of on the scalarisation weight vectors (see the documentation of MOEADr and references therein for details of these methods) as solvers of the hypothetical problem class represented by the available test instances. The performance of each algorithm on each instance will be measured according to an indicator known as *Inverted Generational Distance* (IGD - details here), for which *smaller = better*.

As described in the documentation of both `run_experiment()`

and `calc_nreps()`

, an `algorithm`

must contain an `algorithm$FUN`

field (the name of the function that calls the algorithm) and any other elements/parameters that `algorithm$FUN`

requires (e.g., stop criteria, operator names and parameters, etc.). An additional field, `algorithm$alias`

, can be used to provide the algorithm with a unique identifier.

Supposing that the list in `algorithm`

has fields `algorithm$FUN = myalgo`

, `algorithm$par1 = "a"`

, `algorithm$par2 = 5`

, then the function in `algorithm$FUN`

must have the following structure:

```
myalgo <- function(par1, par2, instance, ...){
# do stuff
return(results)
}
```

That is, it must be able to run if called as:

```
# remove '$FUN' and '$alias' field from list of arguments
# and include the problem definition as field 'instance'
myargs <- algorithm[names(algorithm) != "FUN"]
myargs <- myargs[names(myargs) != "alias"]
myargs$instance <- instance
# call 'algorithm$FUN' with the arguments in 'myargs'
do.call(algorithm$FUN, args = myargs)
```

Finally, the `algorithm$FUN`

routine must return a list object containing (at least) the performance value of the final solution obtained after a given run, in a field named `value`

(e.g., `result$value`

) .

To build the algorithm functions to be used in `run_experiment()`

, we encapsulate (almost) all algorithm parameters within a `myalgo()`

function, which receives only two inputs: the instance to be solved (i.e., one element from `Instance.list`

) and the specification of which version of the algorithm is to be run.

```
# Prepare algorithm function to be used in run_experiment():
myalgo <- function(type, instance){
# Input parameters:
# - type (variant to use: "original", "original2", "moead.de" or "moead.de2")
# - instance (instance to be solved, e.g., instance = instances[[i]])
# All other parameters are set internally
## Extract instance information to build the MOEADr problem format
fdef <- unlist(strsplit(instance$FUN, split = "_"))
uffun <- smoof::makeUFFunction(dimensions = as.numeric(fdef[3]),
id = as.numeric(fdef[2]))
fattr <- attr(uffun, "par.set")
prob.dim <- fattr$pars$x$len
## Build MOEADr problem list
problem <- list(name = instance$FUN,
xmin = fattr$pars$x$lower,
xmax = fattr$pars$x$upper,
m = attr(uffun, "n.objectives"))
## Load presets for the algorithm provided in input 'type' and
## modify whatever is needed for this particular experiment
de2 <- FALSE
if (type == "moead.de2"){
de2 <- TRUE
type <- "moead.de"
}
algo.preset <- MOEADr::preset_moead(type)
algo.preset$decomp$H <- 99 # <-- set population size
algo.preset$stopcrit[[1]]$name <- "maxeval" # <-- type of stop criterion
algo.preset$stopcrit[[1]]$maxeval <- 2000 * prob.dim # stop crit.
poly.ind <- which(sapply(algo.preset$variation,
function(x){x$name == "polymut"}))
algo.preset$variation[[poly.ind]]$pm <- 1 / prob.dim # <--- pm = 1/d
if (de2){
algo.preset$aggfun$name <- "pbi"
algo.preset$aggfun$theta <- 5
algo.preset$neighbors$name = "x"
}
## Run algorithm on "instance"
out <- MOEADr::moead(preset = algo.preset, problem = problem,
showpars = list(show.iters = "none"))
## Read reference data to calculate the IGD
Yref <- as.matrix(read.table(paste0("./inst/extdata/pf_data/",
fdef[1], fdef[2], ".dat")))
IGD = MOEADr::calcIGD(Y = out$Y, Yref = Yref)
## Return IGD as field "value" in the output list
return(list(value = IGD))
}
```

Finally, the `algorithms`

parameter must be assembled as a list of algorithm objects (each containing fields `$FUN`

, `$alias`

and, in this case, `$type`

).

```
# Assemble Algorithm.list. Notice that we need to provide an alias for each
# method, since both algorithms have the same '$FUN' argument.
algorithms <- list(list(FUN = "myalgo",
alias = "Original 1",
type = "original"),
list(FUN = "myalgo",
alias = "Original 2",
type = "original2"),
list(FUN = "myalgo",
alias = "MOEAD-DE",
type = "moead.de"),
list(FUN = "myalgo",
alias = "MOEAD-DE2",
type = "moead.de2"))
```

With the definitions above it is possible now to run an experiment using the iterative sample size determination implemented in CAISEr. For that, all we have to do is define the desired experimental parameters and use `run_experiment()`

.

**IMPORTANT NOTICE:** running this experiment takes a while (about 4 hours in a 3.6 GHz Intel Core i7 iMac with 16Gb RAM, using 7 cores). If you want a (much) faster example, check the documentation of `run_experiment()`

.

```
my.results <- run_experiment(instances = instances,
algorithms = algorithms,
power = 0.8, # Desired power: 80%
power.target = "mean", # on average,
d = 0.5, # to detect differences greater
# than 0.5 standard deviations
sig.level = 0.05, # at a 95% confidence level.
se.max = 0.05, # Measurement error: 5%
dif = "perc", # on the paired percent
# differences of means,
method = "param", # calculated using parametric
# formula.
comparisons = "all.vs.all", # Compare all algorithms
# vs all others,
nstart = 15, # start with 15 runs/algo/inst
nmax = 200, # and do no more than 200 runs/inst.
seed = 1234, # PRNG seed (for reproducibility)
#
# NOTICE: Using all but 1 cores. Change if needed
ncpus = parallel::detectCores() - 1)
```

After that we can interrogate the results and perform inference, if we are so inclined. For instance, we can check if all paired differences in performance are (at least approximately) Normal, so that we can assume a Normal sampling distribution of the means and use t-tests without much worry:

```
suppressPackageStartupMessages(library(car))
algopairs <- paste(my.results$data.summary$Alg1,
my.results$data.summary$Alg2,
sep = " - ")
par(mfrow = c(3, 2))
for (i in seq_along(unique(algopairs))){
tmp <- my.results$data.summary[algopairs == unique(algopairs)[i], ]
car::qqPlot(tmp$Phi,
pch = 16, las = 1, main = unique(algopairs)[i],
ylab = "observed", xlab = "theoretical quantiles")
}
```

The normal QQ plots indicate some deviation from normality, with the `MOEA/D-DE`

vs. `MOEA/D-DE2`

comparison having a particularly longer left tail. We can check visually the effect of this in the sampling distribution of the means:

```
par(mfrow = c(3, 2))
for (i in seq_along(unique(algopairs))){
tmp <- my.results$data.summary[algopairs == unique(algopairs)[i], ]
boot.means <- CAISEr::boot_sdm(tmp$Phi, boot.R = 999)
hist(boot.means, breaks = 30, main = unique(algopairs)[i], las = 1)
}
```

The deviations does not seem to be too severe, but are enough to justify using a nonparametric alternative (e.g., Wilcoxonâ€™s Rank-Sum tests) instead of the t-test. This will sacrifice a little power for borderline cases (i.e., comparisons where the actual effect size is close to the value of \(d^*\) used in the design of the experiment), but give us more confidence in the comparisons that end up being statistically significant.

Some other graphical analysis can be quite informative:

```
df <- cbind(Comparison = algopairs, my.results$data.summary)
suppressPackageStartupMessages(library(ggplot2))
mp <- ggplot(df, aes(x = Comparison, y = Phi, fill = Comparison))
mp +
geom_violin(alpha = 0.6, show.legend = FALSE, scale = "width") +
geom_boxplot(alpha = 0, show.legend = FALSE,
outlier.shape = NA, width = .15) +
geom_point(shape = 16, col = "black", fill = "black", alpha = 0.6,
position = position_jitter(width = .15)) +
geom_abline(slope = 0, intercept = 0, col = "red", lty = 2) +
ylab("Percent difference in IGD") + xlab("") +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
```

Recall first that `smaller = better`

for the quality indicator used in this experiment. Looking at the plot above shows that the differences between all pairs of algorithms were quite high - e.g., differences of up to 200% in the case of the `Original 2`

versus both MOEA/D-DE versions. In general both `MOEA/D-DE`

versions seem to generate better (lower) IGD values than the `Original`

versions, and `MOEA/D-DE`

seems generally better than `MOEA/D-DE2`

(first comparison to the left. Also, the variability seem to be reasonably high for all comparisons.

We can also visualise the individual estimates (with their standard errors):

```
ggplot(df, aes(x = Instance, y = Phi, colour = Comparison,
ymin = Phi - SE, ymax = Phi + SE)) +
geom_pointrange(show.legend = FALSE) +
geom_abline(slope = 0, intercept = 0, col = 1, lty = 2) +
facet_grid(Comparison ~ .) +
theme(axis.text.x = element_text(angle = 60, hjust = 1)) +
xlab("")
```

Notice that the standard errors of the estimators are in most cases smaller than the marker used for the point estimate. Also notice that the observations fall, for each comparison, mainly on one side of the zero-line, which is an almost sure indicator that the comparisons will be statistically significant.

Finally, we can examine our hypothesis tests. The `summary`

method of class `CAISEr`

already outputs the desired information:

```
## #====================================
## CAISEr object:
## Number of instances sampled: 46
## Number of instances required: 46
## Adequate power: TRUE
## Total runs of Original 1 : 1281
## Total runs of Original 2 : 1472
## Total runs of MOEAD-DE : 2020
## Total runs of MOEAD-DE2 : 3317
## #====================================
## Pairwise comparisons of interest:
## Test: wilcoxon
## H1: two.sided
## Comparisons: all.vs.all
## Alpha (FWER): 0.05
## Power target: mean
## Desired power: 0.8
## #====================================
## Tests using Holm's step-down procedure:
##
## Test 1 : Original 1 x Original 2
## H0: median = 0
## alpha = 0.008333
## p-value = 2.842e-14
## Est. median = -0.4038
## CI{1-alpha} = [ -0.5658 -0.2688 ]
## d = -1.310538
##
## Test 2 : MOEAD-DE x MOEAD-DE2
## H0: median = 0
## alpha = 0.01
## p-value = 8.527e-14
## Est. median = -0.1487
## CI{1-alpha} = [ -0.2879 -0.111 ]
## d = -0.9829994
##
## Test 3 : Original 2 x MOEAD-DE
## H0: median = 0
## alpha = 0.0125
## p-value = 4.791e-10
## Est. median = 1.033
## CI{1-alpha} = [ 0.7269 1.47 ]
## d = 1.270841
##
## Test 4 : Original 1 x MOEAD-DE
## H0: median = 0
## alpha = 0.01667
## p-value = 9.02e-06
## Est. median = 0.5934
## CI{1-alpha} = [ 0.2504 0.8769 ]
## d = 0.7954248
##
## Test 5 : Original 2 x MOEAD-DE2
## H0: median = 0
## alpha = 0.025
## p-value = 1.396e-05
## Est. median = 0.7765
## CI{1-alpha} = [ 0.5863 1.194 ]
## d = 0.9214111
##
## Test 6 : Original 1 x MOEAD-DE2
## H0: median = 0
## alpha = 0.05
## p-value = 0.01257
## Est. median = 0.4054
## CI{1-alpha} = [ 0.1109 0.6285 ]
## d = 0.4501527
## #====================================
```

All differences were found to be different at the \(95\%\) joint significance level.

Finally, the full data of the experiment is contained in other fields of the output list `my.results`

, and the user is encouraged to explore these. It is possible, for instance, to can generate box plots and confidence intervals on the mean performance of each algorithm on each sampled instance, which can inspire new questions for the researcher.